一类双曲系统的混沌及其观测器的设计

At present, chaos has become one of the main contents of the nonlinear science, including the complexity of the infinite dimensional dynamical system governed by partial differential equations, such as the turbulence in the famous Navier-Stokes equations.Since the non-compactness of the state space of the PDE systems,it is very difficult to rigorously prove the occurrence of chaos. At present, some simple PDE systems have been well studied, and have obtained some results. The one of the main tasks of this project is to study the chaos of the wave equations with nonlinear boundary condition. Furthermore, we consider the control problem of the system studied in this project. Since the nonlinear term, the classical methods of observer design can not be applied to the system, in this project, we try to give a new method of observer design and rigourously prove the effectiveness of this method, which has many possible applications.

混沌经过几十年的蓬勃发展，已经成为非线性科学的主要研究内容之一，其中偏微分方程所描述的无穷维动力系统的复杂性是目前的研究热点，比如著名的Navier-Stokes方程中的湍流现象。但由于PDE系统状态空间的非紧性，对其研究，特别是要严格证明其具有混沌性态十分困难。目前就某些相对简单的偏微分方程开始了研究，并取得了一些初步的结果。本项目的主要工作之一就是研究带有非线性边界条件的波动方程的混沌性。此外还考虑这类系统的控制问题，由于带有非线性项，经典的观测器设计方法已经失效，本项目尝试提出一种新的观测器设计方法，并给出严格证明，无论是理论意义还是实际背景，都有着广泛的应用前景。

本项目主要研究的是动力系统的混沌性，特别是由PDE所描述的无穷维动力系统的复杂性。此外还研究了一类特殊PDE系统的观测器设计。关于研究成果，具体来说我们严格分析了一类带有混合边界条件波动方程系统的混沌振动，其中在边界条件中含有位移项。此外，我们利用时滞状态反馈的方法构建了一类PDE系统的观测器，并给出了严格的论证和数值模拟。